VOL. XXIX.] I'HILOSOPHICAL TRANSACTIONS. 12/ 



rithm be /; m will be = :; , — - + ■; — 7, — z "~ i — ^ — r~T + &c. If the 



jimui U.V., ^ lX2'lx2x3 1x2x3x4* 



number be greater than unity, as \-\-n, then for finding it, I have likewise 

 discovered the rule, expressed in Mr. Newton's letter; viz. n will be 

 = - -4 1 H — - — - + &c. And as to the regression from 



1 ~1X2^1X2X3 ' 1x2x3x4 ' ^ 



arches, I directly lighted upon the rule that from the given arch gives the 

 co-sine; viz. the co-sine = 1 — j^ + ^^^^^^^ — &c. But afterwards I 

 likewise found, that from this rule might be demonstrated that other 

 communicated to me, for finding the right sine, which is 

 - 1 ^ t . — V — &c. Thus Mr. Leibnitz put in his claim for 



1 1X2X3 ' 1x2x3x4x5 ^ 



the co-invention of these four Series, though the method of finding them was 

 sent him at his own request, and he did not yet understand it. For in this 

 same letter of August 27, 1676, he desired Mr. Newton to explain it further. 

 His words are: " But I wish Mr. Newton would explain some things further ; 

 as, the origin of the Theorem he first lays down ; also the method by which 

 in his operations, he found the quantities />, q, r; and lastly, how he proceeds 

 in the method of regressions, as when, from the logarithm the number is 

 sought. For he does not explain how that is deduced from his method." 



He pretended to have found two Series for the number whose logarithm was 

 given, and yet in the same letter he desired Mr. Newton to explain to him 

 the method of finding those very two Series. 



When Mr. Newton had received this letter, he wrote back that all the said 

 four Series had been communicated by him to Mr. Leibnitz; the first two 

 being one and the same Series in which the letter / was put for the logarithm 

 with its sign -j- or — ; and the third being the excess of the radius above the 

 versed sine, for which a Series had been sent to him. On which Mr. Leibnitz 

 desisted from his claim. Mr. Newton also in the same letter, dated Oct. 24, 

 1676, further explained his methods of regression, as Mr. Leibnitz had desired. 

 And Mr. Leibnitz, in his letter of June 21, l677j desired a further explana- 

 tion : but soon after, on reading Mr. Newton's letter a second time, wrote 

 back July 12, 1677, that he now understood what he wanted; and found by 

 his old papers, that he had formerly used one of Mr. Newton's methods of 

 regression; but in the example which he had then by chance made use of, 

 there being produced nothing elegant, he had, out of his usual impatience, 

 neglected to use it any further. He had therefore several direct Series, and 

 consequently a method of finding them, before he invented and forgot the 

 inverse method. And if he had searched his old papers diligently, he might 

 have found this method also there; but having forgot his own methods, he 

 wrote for Mr. Newton's. 



